notice that ℰG×GX\mathcal{E}G \times_G X is the realization of the action groupoidX//GX//G. This Borel equivariant cohomology theory is what is discussed currently at the entry equivariant cohomology. The following will actually define a refinement of the discussion currently at equivariant cohomology.

We can also see this in the functor of points perspective. Consider the functor Specℤ[t]/(tn)\mathrm{Spec} \; \mathbb{Z} [t]/ (t^n), then for any ring RR $A^1(R)=lim→Specℤ[t]/(tn)(R).\hat \mathbf{A}^1 (R) = \lim_\rightarrow \mathrm{Spec} \; \mathbb{Z} [t]/ (t^n) (R).$ By the universal property of colimits we have

by sending xx to 1+t1 + t, and this corresponds to taking the germ of functions at 1∈Gm1 \in G_m

lesson

given GG an algebraic group such that the formal spectrumSpfA(ℂP∞)Spf A(\mathbb{C}P^\infty) is the completion G^\hat G, define AS1(*):=ℴGA_{S^1}({*}) := \mathcal{o}_{G} then passing to germs gives a completion map